Rainwater Harvesting Model: Units, Roof Area, and Overflow
The garden club wants to collect rainwater from the school roof to irrigate its beds. The building's roof footprint is 1,200 square feet, and the downspout nearest the garden can be routed into 55-gallon rain barrels. A forecast calls for 0.5 inch of rain, and one cubic foot holds 7.48 gallons. A club member calculates: 'Half an inch is 0.5 foot, so 1,200 × 0.5 = 600 cubic feet, about 4,488 gallons - we would need 82 barrels.' Another member argues the figure undercounts, because the pitched roof has more surface area than its footprint and therefore catches more rain. The club has budget for 6 barrels and wants to know whether that is enough for this storm, and how many barrels a 1-inch storm would require.
1. Identify the error in the 4,488-gallon calculation, compute the correct volume in gallons for a 0.5-inch storm, and determine how many 55-gallon barrels are needed, rounding up.
Read the worked answer
The error is a unit conversion. The member used 0.5 foot of rainfall depth when the forecast is 0.5 inch. Since 0.5 in = 0.5/12 = 1/24 ft ≈ 0.0417 ft, the result is 12 times too large.
Correct calculation:
- Depth in feet: 0.5 ÷ 12 = 1/24 ft
- Volume: 1,200 ft² × 1/24 ft = 50 ft³
- Gallons: 50 × 7.48 = 374 gallons
- Barrels: 374 ÷ 55 = 6.8, so 7 barrels to capture the storm completely
With only 6 barrels (330 gallons of capacity), about 44 gallons - roughly 12% of the storm - would overflow and be lost.
Checking the scale makes the error obvious: 4,488 gallons would be nearly a foot and a half of rain over that footprint, not half an inch. A quick sanity check on units (inches must become feet before multiplying by square feet) would have caught it before the club ordered anything.
2. Evaluate the second member's claim that the pitched roof's larger surface area means more water will be captured. Explain what roof property actually governs capture.
Read the worked answer
The claim is incorrect as stated. Rainfall is reported as a depth over horizontal ground, and a tilted plane intercepts exactly the same volume of falling rain as its horizontal projection - the slanted surface spreads the same water over more area rather than collecting more of it. The correct multiplier is the horizontally projected roof area (1,200 ft²), not the slanted surface area, which is larger by 1/cos(pitch angle) but irrelevant to volume.
What does govern capture:
- The projected area that actually drains to this downspout - a 1,200 ft² roof typically sheds to several downspouts, so one barrel system may see only a fraction.
- Rainfall depth.
- Losses: first-flush diversion, gutter splash, evaporation, and overflow once barrels fill.
A second-order exception: strong wind can drive rain onto a windward slope and slightly increase catch, but that is not a planning basis. The member's intuition would only matter if rain were measured along the slope.
3. Write a general expression for gallons captured as a function of rainfall depth r in inches and roof footprint A in square feet. Use it to find the barrels needed for a 1-inch storm, and state one assumption that would make the model overestimate the harvest.
Read the worked answer
Starting from volume in cubic feet and converting to gallons:
G = A × (r/12) × 7.48 = 0.6233 · A · r
where A is the projected area in ft², r is rainfall in inches, and G is gallons.
Check against the 0.5-inch storm: G = 0.6233 × 1,200 × 0.5 = 374 gallons, matching the earlier result.
1-inch storm: G = 0.6233 × 1,200 × 1 = 748 gallons. Barrels = 748 ÷ 55 = 13.6, so 14 barrels.
Assumption that would overestimate the harvest: that the entire 1,200 ft² footprint drains to the barrel-fed downspout. If only half the roof drains there, the 1-inch storm yields about 374 gallons and 7 barrels - half the prediction. Other overestimating assumptions: no first-flush diversion or gutter losses (typically 10–20%), and no overflow once barrels are full.
The model is linear in both A and r, so it scales cleanly - but linearity is only as good as the assumption that all captured area is routed to storage.